Pc Graph Linear Ineq

Graphing Linear Inequalities in Two Variables

Expressions of the type x + 2y ≤ 8 and 3x – y > 6 are called linear inequalities in two variables.

A solution of a linear inequality in two variables is an ordered pair (x, y) which makes the inequality true.

Example: (1, 3) is a solution to x + 2y ≤ 8 since (1) + 2(3) = 7 ≤ 8.

The solution set, or feasible set, of a linear inequality in two variables is the set of all solutions. y

Example: The solution set for x + 2y ≤ 8 is the shaded region.

2 2

The solution set is a half-plane. It consists of the line x + 2y ≤ 8 and all the points below and to its left. The line is called the boundary line of the half-plane.

x

If the inequality is ≤ or ≥ , the boundary line is solid; its points are solutions.

3x – y = 2

y

x

3x – y < 2 Example: The boundary line of the 3x – y > 2 solution set of 3x – y ≥ 2 is solid.

If the inequality is < or >, the boundary line is dotted; its points are not solutions. Example: The boundary line of the solution set of x + y < 2 is dotted.

y

x

A test point can be selected to determine which side of the half-plane to shade. y

Example: For 2x – 3y ≤ 18 graph the boundary line.

(0, 0) -2

2

x

The solution set is a half-plane. Use (0, 0) as a test point. (0, 0) is a solution. So all points on the (0, 0) side of the boundary line are also solutions. Shade above and to the left of the line.

To graph the solution set for a linear inequality: 1. Graph the boundary line. 2. Select a test point, not on the boundary line, and determine if it is a solution. 3. Shade a half-plane.

Example: Graph the solution set for x – y > 2. 1. Graph the boundary line x – y = 2 as a dotted line. y

(0, 0)

2. Select a test point not on the line, say (0, 0).

(2, 0)

x

(0, -2)

(0) – 0 = 0 > 2 is false. 3. Since this is a not a solution, shade in the half-plane not containing (0, 0).

Solution sets for inequalities with only one variable can be graphed in the same way. y

Example: Graph the solution set for x < - 2.

4

x -4

4 -4

y

Example: Graph the solution set for x ≥ 4.

4

x -4

4 -4